arXiv:2507.22787v5 Announce Type: replace-cross 
Abstract: We introduce a three-dimensional vectorial extension of the Hopfield associative-memory model in which each neuron is a unit vector on $S^2$ and synaptic couplings are $3\times 3$ blocks generated through a vectorial Hebbian rule. The resulting block-structured operator is mathematically analogous to the Hessian of amorphous solids and induces a rigid energy landscape with deep minima for stored patterns. Simulations and spectral analysis show that the vectorial network substantially outperforms the classical binary Hopfield model. For moderate connectivity, the critical storage ratio $\gamma_c$ grows approximately linearly with the coordination number $Z$, while for $Z\gtrsim 40$ a high-connectivity regime emerges in which $\gamma_c$ systematically exceeds the extrapolated low-$Z$ linear fit. At the same time, a persistent spectral gap separates pattern modes from the bulk and basins of attraction enlarge, yielding enhanced robustness to initialization noise. Thus geometric constraints combined with amorphous-solid-inspired structure produce associative memories with superior storage and retrieval performance, especially in the high-connectivity ($Z \gtrsim 20$-$30$) regime.

تم تقديم نموذج جديد ثلاثي الأبعاد من نموذج الذاكرة الجمعية هوبفيلد، حيث يتم تمثيل كل خلية عصبية كمتجه وحدة على الكرة S^2. يستخدم هذا النموذج الروابط المشبكية المنظمة في كتل 3x3 من خلال قاعدة هببية متجهة، مما ينتج عنه مشغل هيكلي مشابه لهسيان المواد غير المتبلورة، مما يخلق منظر طاقة صلب مع قيعان عميقة للأنماط المخزنة.

Se ha introducido un nuevo modelo de extensión vectorial tridimensional del modelo de memoria asociativa de Hopfield, donde cada neurona se representa como un vector unitario en la esfera S^2. Este modelo emplea acoplamientos sinápticos organizados en bloques de 3x3 a través de una regla hebbiana vectorial, resultando en un operador estructurado en bloques similar al Hessiano de sólidos amorfos, que crea un paisaje energético rígido con mínimos profundos para los patrones almacenados.

Un nouveau modèle d'extension vectorielle tridimensionnelle du modèle de mémoire associative de Hopfield a été introduit, où chaque neurone est représenté comme un vecteur unitaire sur la sphère S^2. Ce modèle utilise des couplages synaptiques organisés en blocs 3x3 selon une règle d'Hebbian vectorielle, résultant en un opérateur structuré par blocs similaire au Hessien des solides amorphes, créant un paysage énergétique rigide avec des minima profonds pour les motifs stockés.

A new three-dimensional vectorial extension of the Hopfield associative-memory model has been introduced, where each neuron is represented as a unit vector on the sphere S^2. This model employs synaptic couplings organized in 3x3 blocks through a vectorial Hebbian rule, resulting in a block-structured operator similar to the Hessian of amorphous solids, which creates a rigid energy landscape with deep minima for stored patterns.

Amorphous Solid Model of Vectorial Hopfield Neural Networks

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